Mathematics : Continuity and Differentiability

Function f(x) = $$\begin{gathered} \left\{\mathrm{x} - 1, \mathrm{x} <\hspace{0.17em}2 \\ 2\mathrm{x} - 3, \mathrm{x} \ge 2\right\} \end{gathered}$$ is a continuous function

• for x = 2 only

• for all real values of x such that x $\ne$ 2

• for all real values of x

• for all integral values of x only

Differential coefficient of $\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)}$ is

• $\frac{1}{4\sqrt{\mathrm{x}}}\sec \left(\sqrt{\mathrm{x}}\right)\sin \left(\sqrt{\mathrm{x}}\right)$

• $\frac{1}{4\sqrt{\mathrm{x}}}{\left(\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} . \sin \left(\sqrt{\mathrm{x}}\right)$

• $\frac{1}{2}\sqrt{\mathrm{x}}\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)} . \sin \left(\sqrt{\mathrm{x}}\right)$

• $\frac{1}{2}\sqrt{\mathrm{x}}{\left(\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} . \sin \left(\sqrt{\mathrm{x}}\right)$

x = 0, the function f(x) = $\left|\mathrm{x}\right|$ is

• continuous but not differentiable

• discontinuous and differentiable

• discontinuous and not differentiable

• continuous and differentiable

Let f(x) = $$\begin{gathered} \left\{\frac{\sin \left(\mathrm{\pi x}\right)}{5\mathrm{x}}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$if f(x) is continuous at x = 0, then k is equal to

• $\frac{\mathrm{\pi }}{5}$

• $\frac{5}{\mathrm{\pi }}$

• 1

• 0

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\mathrm{x}}\right)\right)$ is equal to :

• $\frac{1}{1 + {\mathrm{x}}^2}$

• $\frac{{\mathrm{x}}^2}{2\sqrt{1 + {\mathrm{x}}^2}\left(\sqrt{1 + {\mathrm{x}}^2} - 1\right)}$

• $\frac{2}{1 + {\mathrm{x}}^2}$

• $\frac{1}{2\left(1 + {\mathrm{x}}^2\right)}$

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\tan }^{-1}\left(\sqrt{\frac{1 + \cos \left(\frac{\mathrm{x}}{2}\right)}{1 - \cos \left({\displaystyle \frac{\mathrm{x}}{2}}\right)}}\right)\right)$ is equal to :

• - 1/4

• 1/4

• - 1/2

• 1/2

If f(x) = x$\left[\sqrt{\mathrm{x}} - \sqrt{\mathrm{x} +\hspace{0.17em}1}\right]$, then :

• f(x) is continuous but not differentiable at x = 0

• f(x) is not differentiable at x = 0

• f(x) is differentiable at x = 0

• None of the above

Let y = t¹⁰ + 1 and x = t⁸ + 1, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to :

• $\frac{5}{2}\mathrm{t}$

• 20t⁸

• $\frac{5}{16{\mathrm{t}}^6}$

• None of these

The vectors $\overrightarrow{\mathrm{AB}} = 3\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 4\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{AC}} = 5\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ are the sides of a triangle ABC. The length ofthe median through A is :

• $\sqrt{13} \mathrm{unit}$

• 2$\sqrt{5}$ unit

• 5 unit

• 10 unit

A function f on R into itself is continuous at a point ain R, iff for each $\in$ > 0, there exists, $\mathrm{\delta }$ > 0 such that :

• $\left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| < \in \Rightarrow \left|\mathrm{x} - \mathrm{a}\right| < \mathrm{\delta }$

• $\left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| > \in \Rightarrow \left|\mathrm{x} - \mathrm{a}\right| > \mathrm{\delta }$

• $\left|\mathrm{x} - \mathrm{a}\right| > \mathrm{\delta } \Rightarrow \left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| > \in$

• $\left|\mathrm{x} - \mathrm{a}\right| < \mathrm{\delta } \Rightarrow \left|\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{a}\right)\right| < \in$